Copyright © 2012 Luis M. Navas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The Bernoulli polynomials restricted to and extended by
periodicity have nth sine and cosine Fourier coefficients of the form . In general, the Fourier coefficients of any polynomial restricted to are
linear combinations of terms of the form . If we can make this linear
combination explicit for a specific family of polynomials, then by uniqueness
of Fourier series, we get a relation between the given family and the Bernoulli
polynomials. Using this idea, we give new and simpler proofs of some known identities
involving Bernoulli, Euler, and Legendre polynomials. The method can also be
applied to certain families of Gegenbauer polynomials. As a result, we obtain
new identities for Bernoulli polynomials and Bernoulli numbers.